Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437452 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,12,8)(7,16,20,28)(9,14,24,21)(10,15,11,22)(13,26,17,19)(18,31,35,36)(23,29,25,30)(27,33,32,34);(1,2)(3,4)(5,7)(6,13)(8,17)(9,18)(10,19)(11,26)(12,20)(14,27)(15,28)(16,22)(21,32)(23,33)(24,35)(25,34)(29,31)(30,36);(2,4)(5,8,12,6)(7,17,20,13)(9,21,24,14)(10,22,11,15)(16,26,28,19)(18,32,35,27)(23,30,25,29)(31,33,36,34);(2,4)(5,9)(6,14)(7,18)(8,21)(10,25)(11,23)(12,24)(13,27)(15,30)(16,31)(17,32)(19,34)(20,35)(22,29)(26,33)(28,36);(5,10,12,11)(6,15,8,22)(7,19,20,26)(9,23,24,25)(13,28,17,16)(14,29,21,30)(18,33,35,34)(27,31,32,36)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72302921, #72303070, #72303150, … |
| maximal normal subgroup of | #72302921, #72303070, #72303150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344655);